Molecular Statistical Theory of the Frank Elastic Constants of Liquid Crystals

The Frank elastic constants K 1, K 2, K 3 are calculated in the mean field approximation by assuming that the intermolecular force is the sum of hard rod repulsion (length L and width D) and Maier-Saupe's type attraction. The main conclusions are as follows. (1) the inequlaities K 3≫ K 1 > K 2 necessarily hold. (2) All the K'is are nearly proportional to the square of the orientational order parameter S. (3) In thermotropic system, K 1/S 2, K 1/K and K 2/K(K: mean elastic constant) increase slowly and K 3/K, K 3/K 1 decreases with L/D, whereas K/kT c is nearly constant for L/D = 3 ∼ 5. These tendecies agree fairly well with the observations on liquid crystals with almost rigid structure. (4) In the lyotorpic system in which L/D ≫ 1, K 1/3 = K 2 = cLk T/3π D 2 and k 1/K 3 = 4D/5cL are obtained (c: packing fraction). The estimate for PBLG solution nearly agrees with recent experimental data.